How to add another option to set object origin hotkey
To adjust an existing menu, you need to start by finding the python class name of the existing menu you want to alter. You then create a function that uses the same argument list as a classes draw method and append it to the menu in bpy.types. You can see an example of this in the add object template script.

In the case of the set origin operator, the menu is created by the operator, it is based on an enum that appears to only exist in the C source code and isn't exposed to python, meaning it doesn't appear to be adjustable,

If you can't find the name of the menu, then you can create a replacement menu yourself. You would then create a shortcut that calls your menu and disable the original shortcut.

An example addon can be found here.